Parity-check matrix of a bar product
= Parity-check matrix of a bar product
If $H_i$ is a <parity-check matrix> of $C_i$, then a parity-check matrix of $C_1|C_2$ is
$$
\begin{pmatrix}H_1&0\\H_2&H_2\end{pmatrix}.
$$
Its equations say that the first half $u$ lies in $C_1$ and that the sum of the two halves lies in $C_2$.